List of logic symbols
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See also: Logical connective
In logic, a set of symbols is commonly used to express logical representation. As logicians are familiar with these
symbols, they are not explained each time they are used. So, for students of logic, the following table lists many
common symbols together with their name, pronunciation, and the related field of mathematics. Additionally, the third
column contains an informal definition, and the fourth column gives a short example.
Be aware that, outside of logic, different symbols have the same meaning, and the same symbol has, depending on the
context, different meanings.
Basic logic symbols
Name
Symbol
Should be
read as
Explanation
Examples
Unicode
Value
HTML
Entity
U+21D2
⇒
U+2192
→
U+2283
⊃
U+21D4
⇔
LaTeX
symbol
Category
material
implication
A ⇒ B is true
just in the case
implies; if .. that either A is
false or B is
then
true, or both.
⇒
→
⊃
→ may mean
the same as ⇒
(the symbol
may also
indicate the
domain and
propositional codomain of a
logic, Heyting function; see
algebra table of
mathematical
symbols).
x = 2 ⇒ x 2 = 4 is true,
but x 2 = 4 ⇒ x = 2 is in
general false (since x
could be −2).
\Rightarrow
\to
\supset
\implies
⊃ may mean
the same as ⇒
(the symbol
may also mean
superset).
⇔
≡
material
equivalence
A ⇔ B is true
if and only if; just in case
iff; means the either both A
same as and B are
false, or both
x + 5 = y + 2 ⇔ x + 3 =
y
\Leftrightarrow
\equiv
U+2261
≡
\leftrightarrow
↔
false, or both
A and B are
propositional
true.
logic
negation
¬
˜
!
∧
not
&
∨
A slash placed ¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y)
propositional through
another
logic
operator is the
same as "¬"
placed in
front.
ǀǀ
⊕
⊻
U+00AC
¬
U+02DC
˜ ~
U+2227
∧
U+0026
&
\wedge or
\land
\&[1]
U+2228
∨
\lor or \vee
⊕
\oplus
\veebar
T
\top
\lnot or \neg
\sim
logical
conjunction
The statement
A ∧ B is true n < 4 ∧ n >2 ⇔ n = 3
propositional if A and B are when n is a natural
logic, both true; else number.
Boolean it is false.
algebra
logical
disjunction
or
+
&harr;
The statement
¬A is true if
and only if A is
false.
and
•
\leftrightarrow
\iff
U+2194
propositional
logic,
Boolean
algebra
exclusive
disjunction
xor
propositional
logic,
Boolean
algebra
The statement
A ∨ B is true
if A or B (or
n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3
both) are true;
when n is a natural
if both are
number.
false, the
statement is
false.
The statement
A ⊕ B is true
when either A
(¬A) ⊕ A is always true,
or B, but not
A ⊕ A is always false.
both, are true.
A ⊻ B means
the same.
U+2295
U+22BB
Tautology
⊤
T
top, verum
The statement
⊤ is
A ⇒ ⊤ is always true.
unconditionally
propositional
logic, true.
Boolean
U+22A4
Boolean
algebra
1
⊥
F
0
∀
()
∃
∃!
Contradiction
bottom,
falsum
The statement
⊥ is
propositional unconditionally ⊥ ⇒ A is always true.
logic, false.
Boolean
algebra
universal
quantification ∀ x: P(x) or
for all; for (x) P(x)
∀ n ∈ ℕ: n2 ≥ n.
any; for each means P(x) is
first­order true for all x.
U+22A5
&perp;
F
\bot
U+2200
&forall;
\forall
U+2203
&exist;
\exists
logic
existential
∃ x: P(x)
quantification means there is
there exists at least one x ∃ n ∈ ℕ: n is even.
first­order such that P(x)
logic is true.
uniqueness
quantification ∃! x: P(x)
means there is
there exists
exactly one x ∃! n ∈ ℕ: n + 5 = 2n.
exactly one
such that P(x)
first­order is true.
logic
\exists !
definition
:=
≡
:⇔
x := y or x ≡ y
means x is
defined to be
is defined as another name
for y (but note
that ≡ can also cosh x := (1/2)(exp x +
U+2203 U+0021 &exist; !
mean other
exp (−x))
things, such as
congruence). A XOR B :⇔
(A ∨ B) ∧ ¬(A ∧ B)
everywhere P :⇔ Q means
P is defined to
be logically
equivalent to
Q.
U+2254
:=
(U+003A U+003D)
:
U+2261
&equiv;
:=
\equiv
\Leftrightarrow
U+003A U+229C
&hArr;
Q.
( )
precedence
grouping
Perform the
operations
parentheses, inside the
brackets parentheses
everywhere first.
(8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1,
but 8 ÷ (4 ÷ 2) = 8 ÷ 2 =
4.
U+0028 U+0029
( )
( )
U+22A2
&#8866;
\vdash
U+22A8
&#8872;
\models
Turnstile
⊢
x ⊢ y means
provable y is provable
from x (in
propositional some specified A → B ⊢ ¬B → ¬A
logic, first­ formal
order logic system).
double
turnstile
⊨
x ⊨ y means
x semantically A → B ⊨ ¬B → ¬A
propositional entails y
logic, first­
order logic
entails
Advanced and rarely used logical symbols
These symbols are sorted by their Unicode value:
U+00B7 ∙ MIDDLE DOT , an outdated way for denoting AND[citation needed], still in use in electronics; for
example "A∙B" is the same as "A&B"
∙: Center dot with a line above it. Outdated way for denoting NAND, for example "A∙B" is the same as "A
NAND B" or "A|B" or "¬(A & B)". See also Unicode U+22C5 ⋅ DOT OPERAT OR.
U+0305 ̅
COMBINING OVERLINE, used as abbreviation for standard numerals. For example, using HTML style
"4̅
" is a shorthand for the standard numeral "SSSS0".
Overline, is also a rarely used format for denoting Gödel numbers, for example "AVB" says the Gödel number
of "(AVB)"
Overline is also an outdated way for denoting negation, still in use in electronics; for example "AVB" is the same
as "¬(AVB)"
U+2191 ↑ UPWARDS ARROW or U+007C | VERT ICAL LINE: Sheffer stroke, the sign for the NAND operator.
U+2201 ∁ COMPLEMENT
U+2204 ∄ T HERE DOES NOT EXIST : strike out existential quantifier same as "¬∃"
U+2234 ∴ T HEREFORE
U+2235 ∵ BECAUSE
U+22A7 ⊧ MODELS: is a model of
U+22A8 ⊨ T RUE: is true of
U+22AC ⊬ DOES NOT PROVE: negated ⊢, the sign for "does not prove", for example T ⊬ P says "P is not a
theorem of T"
U+22AD ⊭ NOT T RUE: is not true of
U+22BC ⊼ NAND: another NAND operator, can also be rendered as ∧
U+22BD ⊽ NOR: another NOR operator, can also be rendered as V
U+22C4 ⋄ DIAMOND OPERAT OR: modal operator for "it is possible that", "it is not necessarily not" or rarely "it is
not provable not" (in most modal logics it is defined as "¬◻¬")
U+22C6 ⋆ ST AR OPERAT OR: usually used for ad­hoc operators
U+22A5 ⊥ UP T ACK or U+2193 ↓ DOWNWARDS ARROW : Webb­operator or Peirce arrow, the sign for NOR.
Confusingly, "⊥" is also the sign for contradiction or absurdity.
U+2310 ⌐ REVERSED NOT SIGN
U+231C ⌜ T OP LEFT CORNER and U+231D ⌝ T OP RIGHT CORNER: corner quotes, also called "Quine quotes"; for
quasi­quotation, i.e. quoting specific context of unspecified ("variable") expressions;[2] also the standard
symbol[citation needed] used for denoting Gödel number; for example "⌜G⌝" denotes the Gödel number of G.
(Typographical note: although the quotes appears as a "pair" in unicode (231C and 231D), they are not
symmetrical in some fonts. And in some fonts (for example Arial) they are only symmetrical in certain sizes.
Alternatively the quotes can be rendered as ⌈ and ⌉ (U+2308 and U+2309) or by using a negation symbol and
a reversed negation symbol ⌐ ¬ in superscript mode. )
U+25FB ◻ WHIT E MEDIUM SQUARE or U+25A1 □ WHIT E SQUARE: modal operator for "it is necessary that" (in
modal logic), or "it is provable that" (in provability logic), or "it is obligatory that" (in deontic logic), or "it is
believed that" (in doxastic logic).
Note that the following operators are rarely supported by natively installed fonts. If you wish to use these in a web
page, you should always embed the necessary fonts so the page viewer can see the web page without having the
necessary fonts installed in their computer.
U+27E1 ⟡ WHIT E CONCAVE­ SIDED DIAMOND
U+27E2 ⟢ WHIT E CONCAVE­ SIDED DIAMOND WIT H LEFT WARDS T ICK: modal operator for was never
U+27E3 ⟣ WHIT E CONCAVE­ SIDED DIAMOND WIT H RIGHT WARDS T ICK: modal operator for will never be
U+27E4 ⟤ WHIT E SQUARE WIT H LEFT WARDS T ICK: modal operator for was always
U+27E5 ⟥ WHIT E SQUARE WIT H RIGHT WARDS T ICK: modal operator for will always be
U+297D ⥽ RIGHT FISH T AIL: sometimes used for "relation", also used for denoting various ad hoc relations (for
example, for denoting "witnessing" in the context of Rosser's trick) The fish hook is also used as strict
implication by C.I.Lewis ⥽ , the corresponding LaTeX macro is \strictif. See here
(http://www.fileformat.info/info/unicode/char/297d/index.htm) for an image of glyph. Added to Unicode 3.2.0.
See also
Logic Alphabet, a suggested set of logical symbols
Mathematical operators and symbols in Unicode
Polish notation
List of mathematical symbols
Notes
1. ^ Although this character is available in LaTeX, the MediaWiki TeX system doesn't support this character.
2. ^ Quine, W.V. (1981): Mathematical Logic, §6
External links
Named character entities (http://www.w3.org/TR/WD­html40­970708/sgml/entities.html) in HTML 4.0
Retrieved from "http://en.wikipedia.org/w/index.php?title=List_of_logic_symbols&oldid=589580220"
Categories: Mathematical notation Logic symbols
This page was last modified on 7 January 2014 at 10:05.
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